Using the given information, the height of the shorter cone is 6.81 cm
Calculating volume of a cone
From the question, we are to determine the height of the shorter cone.
First, we will calculate the volume of the first cone
Using the formula for calculating the volume of a cone,
[tex]V = \frac{1}{3}\pi r^{2}h[/tex]
Where V is the volume
r is the radius
and h is the height
From the given information,
r = 8.5 cm
h = 10 cm
Then,
[tex]V = \frac{1}{3}\pi \times 8.5^{2}\times 10[/tex]
[tex]V = \frac{1}{3}\pi \times 72.25\times 10[/tex]
[tex]V = \frac{722.5}{3}\pi[/tex] cm³
Now, for the shorter cone
r = 10.3 cm
Since the two cones have the exact same volume, we can write that
[tex]\frac{722.5}{3}\pi= \frac{1}{3}\pi \times 10.3^{2} \times h[/tex]
[tex]722.5= 106.09 \times h[/tex]
[tex]h = \frac{722.5}{106.09}[/tex]
h = 6.81 cm
Hence, the height of the shorter cone is 6.81 cm.
Learn more on Calculating volume of a cone here: https://brainly.com/question/12004994
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